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If
int_(1)^(8)[3f(x)+2]dx=29, find
int_(1)^(8)f(x)dx

If $\int_{1}^{8}[3 f(x)+2] d x=29$ , find $\int_{1}^{8} f(x) d x$

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Evaluate definite integrals using the chain rule

Full solution

Q. If

\int_{1}^{8}[3 f(x)+2] d x=29

, find

\int_{1}^{8} f(x) d x

Separate into Two Parts: We are given the integral of a function plus a constant: $\int_{1}^{8}[3f(x)+2]\,dx=29$ . We can separate this integral into two parts: the integral of $3f(x)$ from $1$ to $8$ and the integral of $2$ from $1$ to $8$ .
Integral of Constant $2$ : First, let's find the integral of the constant $2$ from $1$ to $8$ . The integral of a constant $a$ from $b$ to $c$ is $a(c - b)$ . So, the integral of $2$ from $1$ to $8$ is $1$ $0$ .
Subtract Constant Integral: Now, we subtract the integral of the constant from the given integral to find the integral of $3f(x)$ from $1$ to $8$ . We have $29$ (the given integral) minus $14$ (the integral of the constant), which equals $15$ .
Find Integral of $3f(x)$ : The integral of $3f(x)$ from $1$ to $8$ is $3$ times the integral of $f(x)$ from $1$ to $8$ . So, if $3$ times the integral of $f(x)$ from $1$ to $8$ equals $3f(x)$ $2$ , then the integral of $f(x)$ from $1$ to $8$ is $3f(x)$ $2$ divided by $3$ .
Divide by $3$ : Dividing $15$ by $3$ gives us the integral of $f(x)$ from $1$ to $8$ , which is $5$ .

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